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Theorem dcomex 10442
Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)
Assertion
Ref Expression
dcomex ω ∈ V

Proof of Theorem dcomex
Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 8488 . . . . . . 7 1o ≠ ∅
2 df-br 5150 . . . . . . . 8 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩})
3 elsni 4646 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩)
4 fvex 6905 . . . . . . . . . 10 (𝑓𝑛) ∈ V
5 fvex 6905 . . . . . . . . . 10 (𝑓‘suc 𝑛) ∈ V
64, 5opth1 5476 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩ → (𝑓𝑛) = 1o)
73, 6syl 17 . . . . . . . 8 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → (𝑓𝑛) = 1o)
82, 7sylbi 216 . . . . . . 7 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → (𝑓𝑛) = 1o)
9 tz6.12i 6920 . . . . . . 7 (1o ≠ ∅ → ((𝑓𝑛) = 1o𝑛𝑓1o))
101, 8, 9mpsyl 68 . . . . . 6 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1o)
11 vex 3479 . . . . . . 7 𝑛 ∈ V
12 1oex 8476 . . . . . . 7 1o ∈ V
1311, 12breldm 5909 . . . . . 6 (𝑛𝑓1o𝑛 ∈ dom 𝑓)
1410, 13syl 17 . . . . 5 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
1514ralimi 3084 . . . 4 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16 dfss3 3971 . . . 4 (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
1715, 16sylibr 233 . . 3 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18 vex 3479 . . . . 5 𝑓 ∈ V
1918dmex 7902 . . . 4 dom 𝑓 ∈ V
2019ssex 5322 . . 3 (ω ⊆ dom 𝑓 → ω ∈ V)
2117, 20syl 17 . 2 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22 snex 5432 . . 3 {⟨1o, 1o⟩} ∈ V
2312, 12fvsn 7179 . . . . . . . 8 ({⟨1o, 1o⟩}‘1o) = 1o
2412, 12funsn 6602 . . . . . . . . 9 Fun {⟨1o, 1o⟩}
2512snid 4665 . . . . . . . . . 10 1o ∈ {1o}
2612dmsnop 6216 . . . . . . . . . 10 dom {⟨1o, 1o⟩} = {1o}
2725, 26eleqtrri 2833 . . . . . . . . 9 1o ∈ dom {⟨1o, 1o⟩}
28 funbrfvb 6947 . . . . . . . . 9 ((Fun {⟨1o, 1o⟩} ∧ 1o ∈ dom {⟨1o, 1o⟩}) → (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o))
2924, 27, 28mp2an 691 . . . . . . . 8 (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o)
3023, 29mpbi 229 . . . . . . 7 1o{⟨1o, 1o⟩}1o
31 breq12 5154 . . . . . . . 8 ((𝑠 = 1o𝑡 = 1o) → (𝑠{⟨1o, 1o⟩}𝑡 ↔ 1o{⟨1o, 1o⟩}1o))
3212, 12, 31spc2ev 3598 . . . . . . 7 (1o{⟨1o, 1o⟩}1o → ∃𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡)
3330, 32ax-mp 5 . . . . . 6 𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡
34 breq 5151 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → (𝑠𝑥𝑡𝑠{⟨1o, 1o⟩}𝑡))
35342exbidv 1928 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → (∃𝑠𝑡 𝑠𝑥𝑡 ↔ ∃𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡))
3633, 35mpbiri 258 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → ∃𝑠𝑡 𝑠𝑥𝑡)
37 ssid 4005 . . . . . . 7 {1o} ⊆ {1o}
3812rnsnop 6224 . . . . . . 7 ran {⟨1o, 1o⟩} = {1o}
3937, 38, 263sstr4i 4026 . . . . . 6 ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}
40 rneq 5936 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 = ran {⟨1o, 1o⟩})
41 dmeq 5904 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → dom 𝑥 = dom {⟨1o, 1o⟩})
4240, 41sseq12d 4016 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}))
4339, 42mpbiri 258 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 ⊆ dom 𝑥)
44 pm5.5 362 . . . . 5 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
4536, 43, 44syl2anc 585 . . . 4 (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
46 breq 5151 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4746ralbidv 3178 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4847exbidv 1925 . . . 4 (𝑥 = {⟨1o, 1o⟩} → (∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4945, 48bitrd 279 . . 3 (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
50 ax-dc 10441 . . 3 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
5122, 49, 50vtocl 3550 . 2 𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)
5221, 51exlimiiv 1935 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2941  wral 3062  Vcvv 3475  wss 3949  c0 4323  {csn 4629  cop 4635   class class class wbr 5149  dom cdm 5677  ran crn 5678  suc csuc 6367  Fun wfun 6538  cfv 6544  ωcom 7855  1oc1o 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-dc 10441
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-1o 8466
This theorem is referenced by:  axdc2lem  10443  axdc3lem  10445  axdc4lem  10450  axcclem  10452  precsexlem10  27662
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